Anomalous transport independent of gauge fields
AAnomalous transport independent of gauge fields
Mamiya Kawaguchi ∗ and Ken Kikuchi † Department of Physics and Center for Field Theory and Particle Physics,Fudan University, 200433 Shanghai, ChinaSeptember 1, 2020
Abstract
We show that three-dimensional trace anomalies lead to new universal anomaloustransport effects on a conformally flat space-time with background scalar fields. In con-trast to conventional anomalous transports in quantum chromodynamics or quantumelectrodynamics, our current is independent of background gauge fields. Therefore, ouranomalous transport survives even in the absence of vectorlike external sources. Bymanipulating background fields, we suggest a setup to detect our anomalous transport.If one turns on scalar couplings in a finite interval and considers a conformal factordepending just on (conformal) time, we find anomalous transport localized at the inter-faces of the interval flows perpendicularly to the interval. The magnitude of the currentsis the same on the two interfaces but with opposite directions. Without the assumptionon scalar couplings, and only assuming the conformal factor depending solely on (con-formal) time as usually done in cosmology, one also finds the three-dimensional Hubbleparameter naturally appears in our current.
Introduction. − Quantum field theories (QFTs) often have anomalies in global symme-tries, called ’t Hooft anomalies [1], which characterize QFTs. The global symmetries canbe either internal or space-time symmetries. In the latter case, one can still employ theanomalies to study QFTs. A remarkable example is the a theorem [2]. In those papers,they coupled a theory to a background metric specified by a “dilaton.” Some terms of thedilaton effective action survive even after taking the flat limit, which were used to show the a theorem in four-dimensional QFTs defined on the Minkowski space. An important lessonone can learn from this example is that anomalies can have remnants even after turning offbackground fields introduced to diagnose anomalies. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] A ug nomalies also lead to interesting phenomena by inducing currents. Particularly, theanomalous transport is generated by the chiral anomaly in hot and/or dense environments.For example, the chiral separation effect [3, 4] and the chiral magnetic effect [5, 6] induce theaxial current j A and the vector current j V parallel to an external magnetic field B in thedense medium µ V (cid:54) = 0 and the chiral imbalance medium µ A (cid:54) = 0, respectively: (cid:104) j A (cid:105) = µ V π e B , (cid:104) j V (cid:105) = µ A π e B , (1)where µ V and µ A are the ordinary and chiral chemical potentials, respectively. Furthermore,in rotating systems, an additional term shows up in the expression of the axial and vectorcurrent, which is called the chiral vortical effect [7, 8, 9], (cid:104) j A (cid:105) = (cid:18) T µ V + µ A π (cid:19) Ω , (cid:104) j V (cid:105) = µ V µ A π Ω , (2)where Ω denotes the angular velocity. This is not the usual background gauge field, but avectorlike external source, which is not dynamical but rather specified by hand. Besides thechiral anomaly, the anomalous breaking of the scale symmetry also leads to an anomaloustransport. As was discussed in [10], an electric current in a curved space-time is generatedby the scale electric effect and the scale magnetic effect (cid:104) j V (cid:105) = − β ( e ) e ∂τ ( x ) ∂t E ( x ) + 2 β ( e ) e ∇ τ ( x ) × B ( x ) , (3)where τ ( x ) represents the local scale factor and β ( e ) is the beta function of quantum elec-trodynamics (QED).The induced anomalous current at zero temperature completely relies on the backgroundgauge fields as was shown in Eqs. (1)-(3). Therefore, as background gauge fields are turnedoff, the conventional anomalous currents vanish (at zero temperature). [The chiral vorticaleffect (2) at nonzero temperature also vanishes if one turns off the vectorlike external source Ω .] So, as in the proof of the a theorem, if one is eventually interested in A = 0 (and Ω = 0), all known anomalous transports become trivial. In a sense, this conclusion is naturalbecause a current is a space-time vector, and it seems we need a “seed” to produce such avector-valued quantity. Of course, if one looks at broader currents, one can have nonzerocurrents in the absence of vectorlike external sources as in scalar QED. (In that case, one has j µ ∼ φ ∗ ∂ µ φ − φ∂ µ φ ∗ with complex dynamical scalar field φ .) However, if one restricts oneselfto currents induced by anomalies, all known examples vanish once vectorlike external sourcesare turned off. [The conclusion is unaffected even if one takes mass terms into account in(four-dimensional) quantum chromodynamics (QCD) and QED. See Ref. [20] below.] So itis natural to ask “is it really necessary to keep background gauge fields (or more precisely,vectorlike external sources) to have nonzero anomalous transports?” We would like to addressthis question in this paper. And of course the answer is no. One does not have to keep nonzerobackground gauge fields to have nonzero anomalous transports. We construct an example.2 etup. − To concretely derive the anomalous transport independent of external gaugefields, we begin by introducing three-dimensional [11] conformal field theories (CFTs) witha global symmetry G (charge conjugation is not necessarily assumed) on a conformally flatspace-time γ µν ( x ) = e τ ( x ) η µν , (4)where µ = 0 , , η µν = diag(+1 , − , −
1) is the Lorentzian metric, and | τ | (cid:28)
1. Thenvia a conformal map, one can translate results on the geometry (4) to those on the flatspace-time. On considering renormalization group on such a curved space-time, one has touse local renormalization group (LRG) [12, 13]. The LRG transformations are realized byWeyl transformations γ µν ( x ) (cid:55)→ e σ ( x ) γ µν ( x ) . To keep effective action invariant under thetransformations, one has to promote all coupling constants λ I to background scalar fields λ I ( x ), and modify them appropriately according to beta functionals. After introducing thebackground fields, the partition function Z becomes a functional of background fields [14] Z [ τ, λ, A ] := (cid:90) D X exp (cid:110) iS CFT [ X ] + i (cid:90) d xτ ( x ) T µµ ( x ) + i (cid:90) d xλ I ( x ) O I ( x ) − i (cid:90) d xA aµ ( x ) j aµ ( x )+ · · · (cid:111) , (5)where X collectively denotes dynamical fields, A aµ ( x ) is the background G gauge field, and λ I ( x ) denote the coupling constants promoted to the space-time-dependent background scalarfields. The background scalar fields belong to representations R I : G → End( V dim R I R I ( G ) ) of G . One then imposes the generating functional be invariant under LRG operator ∆ σ := (cid:82) d xσ ( x ) { γ µν ( x ) δδγ µν ( x ) + β I [ λ ( x )] δδλ I ( x ) + β aµ [ λ ( x )] δδA aµ ( x ) } up to anomalies, where β I [ λ ] and β aµ [ λ ] ≡ D µ λ I ρ aI [ λ ] are scalar and vector beta functionals, respectively. Namely, one requires∆ σ Z [ τ, λ, A ] = 0 + (anomaly) corresponding to the invariance of observables under a changeof an artificial mass scale µ , e.g., dZd ln µ = 0 in the usual RG.By using the generating functional (5), one can compute expectation values of operatorsincluding trace of the energy-momentum tensor T µµ ( x ), composite (scalar) operators O I ( x ),and currents j aµ ( x ) by taking functional derivatives with respect to appropriate backgroundfields: (cid:104) T µµ ( x ) (cid:105) = − i δ ln Z [ τ, λ, A ] δτ ( x ) , (cid:104) O I ( x ) (cid:105) = − i δ ln Z [ τ, λ, A ] δλ I ( x ) , (cid:104) j aµ ( x ) (cid:105) = i δ ln Z [ τ, λ, A ] δA aµ ( x ) . (6)In CFTs, trace of the energy-momentum tensor vanishes as an operator, however, cansuffer from anomalies [15]. In three dimensions, the general form of the trace anomaly isgiven by [16] (cid:104) T µµ (cid:105) = (cid:15) µνρ C IJK [ λ ] D µ λ I D ν λ J D ρ λ K + (cid:15) µνρ C aI [ λ ] F aµν D ρ λ I , (7)3here (cid:15) µνρ is the Levi-Civita tensor with (cid:15) = +1, F aµν is the field strength of A , i.e., F = dA + [ A, A ], and the covariant derivative is defined by D µ λ I = [ δ IJ ∂ µ + iA aµ ( T a ) IJ ] λ J with T a the representation matrices of R I . Since (cid:15) µνρ is completely antisymmetric, C IJK [ λ ] is alsoa completely antisymmetric three-tensor. Once a CFT is specified, the coefficient functionals C IJK and C aI are fixed. These are analogs of central charges, which in general depend oncouplings. Similarly, “central charges” C IJK and C aI depend on couplings in general. Sinceone promotes coupling constants to background (scalar) fields in LRG, the central chargesare functionals of the background scalar fields in the LRG framework. The functionals aresubject to constraints [16] 3 β I [ λ ] C IJK [ λ ] + ρ aI [ λ ] C aJ [ λ ] − ρ aJ [ λ ] C aI [ λ ] = 0 , β I [ λ ] C aI [ λ ] = 0 , originating from the Wess-Zumino consistency condition [17].The possibility of three-dimensional trace anomaly was studied holographically [18], andconcluded that the anomaly does not exist in odd dimensions if gravity duals of the CFTsexist. However, the trace anomaly has not been ruled out if CFTs do not admit holographicduals. So in the first part of this paper where we impose conformal symmetry, we simply as-sume an existence of the three-dimensional trace anomaly, and later we relax the assumptionof conformal symmetry in which case we do have an example of nonzero trace “anomaly.”The current (cid:104) j aµ (cid:105) induced by the weak background gauge field A aµ ( x ), the small localdilatations of the metric τ ( x ), and the small space-time dependent coupling constants λ I ( x )can be expanded in series of these background fields: (cid:104) j aµ (cid:105) = (cid:104) j aµ (cid:105) τ + (cid:104) j aµ (cid:105) λ + (cid:104) j aµ (cid:105) A + (cid:104) j aµ (cid:105) ττ + (cid:104) j aµ (cid:105) τλ + (cid:104) j aµ (cid:105) τA + (cid:104) j aµ (cid:105) λλ + (cid:104) j aµ (cid:105) λA + (cid:104) j aµ (cid:105) AA + · · · . (8)To perform this expansion systematically, we assume the background fields τ, λ, and A areall proportional to a small (positive) number ε , | ε | (cid:28)
1. The first term (cid:104) j aµ (cid:105) τ correspondsto the linear response of the current to the operator τ ( x ) T µµ ( x ), (cid:104) j aµ ( x ) (cid:105) τ = i (cid:90) d y (cid:104) j aµ ( x ) T νν ( y ) (cid:105) τ ( y ) , (9)where (cid:104)· · ·(cid:105) indicates that the expectation value is evaluated by setting background fieldsto zero after the computation. The correlation function can be calculated easily by takinga functional derivative of (7) with respect to the background gauge field: (cid:104) j aµ ( x ) T νν ( y ) (cid:105) = i δ (cid:104) T νν ( y ) (cid:105) δA aµ ( x ) (cid:12)(cid:12)(cid:12) τ,λ,A → = 0 . Thus the first term of (8) vanishes: (cid:104) j aµ (cid:105) τ = 0. The other terms can becomputed in the same way. Generically nonvanishing terms are given by (cid:104) j aµ (cid:105) λ , (cid:104) j aµ ( x ) (cid:105) A , (cid:104) j aµ ( x ) (cid:105) λλ , (cid:104) j aµ ( x ) (cid:105) λA , (cid:104) j aµ ( x ) (cid:105) AA , and (cid:104) j aµ ( x ) (cid:105) τλ = − (cid:90) d y (cid:90) d z (cid:104) j aµ ( x ) T νν ( y ) O I ( z ) (cid:105) τ ( y ) λ I ( z )= − (cid:90) d y (cid:90) d z δ (cid:104) T νν ( y ) (cid:105) δA aµ ( x ) δλ I ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ,λ,A → τ ( y ) λ I ( z )= 2 (cid:15) µνρ ∂ ν λ I ( x ) ∂ ρ { C aI [ λ ( x )] τ ( x ) } (10)4hese are our main results [19]. [Throughout this paper, we consider generic G , but if oneconsiders G = U (1), one only has to drop adjoint indices a, b, . . . = 1 , , . . . , dim G fromour formulas.] As we mentioned in the Introduction, anomalous currents (10) remain finiteeven after turning off the background gauge field, A = 0. One may ask how nontrivialthis fact is. Actually, as one can show, one cannot obtain nonzero anomalous transports in(four-dimensional) QCD or QED at A = 0 [20]. Therefore, once vectorlike external sourcesare switched off, observed anomalous transports can be identified with our current almostuniquely. Remarkably, the anomalous current with this striking property allows us to havea chance to detect the trance anomaly (7), as explained below.Since expectation values of currents may include λ ’s (possibly with covariant derivatives),generically nonzero correlators listed above (10) are interesting to study on their own. How-ever, since we would like to focus on the anomalous transport induced by the trace anomalyin this paper, we just consider (10) from now on. Note that C aI depends on a theory (namely,if one specifies a CFT, the functional is determined), but the form of the current is universalamong CFTs with the trace anomaly (7). Said differently again, the conventional anoma-lous transport goes away with vanishing vectorlike external sources A = 0, so that it wouldstrongly signal an existence of the trace anomaly (7) if one finds this sort of current. Notethat the local counterterm, (cid:82) d xτ ( x ) λ I ( x ) J I ( x ), induces a current similar to our result (10).However, the induced current has order O ( ε ) if J I depends on A . Thus up to O ( ε ) thecurrent induced by the local counterterm can be dropped. Therefore, our result is intact upto O ( ε ) even if the local counterterm is included. Detecting our anomalous current. − Now, since we found our anomalous current (10) isphysical, we would like to discuss its consequences and how to detect it. All observables onehas in CFTs are correlation functions because there is no asymptotic state. Therefore weshould study correlators including the current operator j aµ . Then we find anomalous currentmanifests in contact terms of higher point functions of j aµ and scalar operators O I or trace ofthe energy-momentum operator T µµ . For example, the anomalous current leads to a contactterm (cid:104) j aµ ( x ) T νν ( y ) (cid:105) (cid:51) − i(cid:15) µνρ ∂ ν λ I ( x ) ∂ xρ { C aI [ λ ( x )] δ (3) ( x − y ) } . In a realistic experiment, it isdifficult to respect the exact conformal symmetry (due to finite size of materials, for example).So as long as we assume conformal symmetry, the only hope to detect the contact term isin lattice simulations. In principle, we believe that the contact term can be detected innumerical lattice simulations. However, since contact terms are sensitive to regularizations,it would be difficult to detect the contact term in lattice simulations. Therefore, in order todiscuss a way to detect our current in a more realistic experiment, we would like to relax theassumption of the conformal symmetry.Without conformal symmetry, one can no longer bring results back to (conformally) flatspace-times. However, as we mentioned in Ref. [19], if the current (10) cannot be shifted withlocal counterterms, it is still physical. Therefore, the remnant of the “anomalous” current[21] can be observed in experiments. Furthermore, because of smaller space-time symmetry,we do have a theory [18] with nonzero trace anomaly (7). The nonzero central charge C aI γ ab = − ρ cI [ λ ] δ bc ( iT a λ ) I of the currentoperator J aµ , where we used the operator equation D µ J aµ = − ( iT a λ ) I O I [see Eq. (43) ofthe paper]. The holography (assumed in the paper) is nowadays applied to condensed matterphysics under the name AdS/CMT correspondence, and there also exist some experimentalchecks (see, say, [22] for a review). Therefore employing the experimental setup, we believeour anomalous current is detectable in real materials at least in theories studied in [18].How do the anomalous currents manifest in experiments? Let us discuss the consequences,and a way to detect them. We propose two places where our anomalous current gives physicalconsequences; one is real materials and another is the boundary of our observable universe.(“Confinement” of degrees of freedom to such boundaries can be realized by some knownmechanisms such as anomaly inflow. However, since we take the effective field theoreticalapproach, we do not care how exactly the situations are realized.) Let us start from a generalconsideration. In the case of G = U (1) and assuming an existence of massless photon field a , one can couple the anomalous current to the dynamical-photon field embedded in A as A = a + ¯ A , where we interpret ¯ A as a background U (1) gauge field which is sent to zero in oursetup. Then the anomalous current induces the single-photon production. The amplitudeof the single-photon production tagged with the momentum q and the polarization (cid:15) ( i ) ( q ) isobtained from the generating function, i M τλ ( i ; q ) = (cid:104) (cid:15) ( i ) ( q ) | (cid:105) = − i (cid:15) ( i ) µ ( q ) (cid:112) (2 π ) q (cid:90) d xe iq · x (cid:104) j µ ( x ) (cid:105) τλ . (11)This amplitude implies that the real photon is surely emitted from the nonzero anomaloustransport. Interestingly, the photon emission (11) is certainly produced even in the absenceof vectorlike external sources in contrast to the single-photon production driven by the con-ventional anomalous transport [23]. So if one detects photons in the absence of backgroundgauge fields, the result suggests the photons are emitted from our anomalous current andnot others. In particular, by manipulating background fields (possible in experiments withreal materials but difficult at the boundary of our observable universe), one can distinguishour current from the others in this way. Furthermore, if photons are detected in the situa-tion, it suggests an experimental discovery of the trace anomaly, which has not been foundtheoretically yet.In order to make the prediction (11) more precise, we would like to manipulate backgroundfields. We take them as we wish in accord with defining properties such as | τ | (cid:28)
1. (Thefollowing discussion can be repeated assuming conformal symmetry.)As a first example, let us set λ I ( x ) = ε I θ ( x ) θ ( l − x ) where ε I ’s are numbers of order ε and l is some length. This choice is motivated by doping in condensed matter physics; onecan dope a material with a fermion ψ only in an interval x ∈ [0 , l ]. Then an example ofposition-dependent scalar coupling is given by a Yukawa coupling λφ ¯ ψψ between ψ and ascalar field φ with λ ∼ θ ( x ) θ ( l − x ). Preparing more Yukawa couplings λ I , we can model6he situation as λ I ’s are turned on only in the interval, λ I ( x ) = ε I θ ( x ) θ ( l − x ). Adoptingthis choice of scalar background fields, we get (cid:104) j a ( x ) (cid:105) τλ = 2 ε I C aI ( ε/ ∂ τ ( x ) (cid:2) δ ( x ) − δ ( l − x ) (cid:3) , (cid:104) j a ( x ) (cid:105) τλ = 0 , (cid:104) j a ( x ) (cid:105) τλ = − ε I C aI ( ε/ ∂ τ ( x ) (cid:2) δ ( x ) − δ ( l − x ) (cid:3) . (12)We would like to make three comments on this result: (i) This result shows that (ifone prepares doped materials as we described) our anomalous current, perpendicular to theinterval x ∈ [0 , l ], is generated at the interface of doped intervals. (ii) The form of anomalouscurrent is independent of models (only central charges C aI depend on theories), and currentnecessarily exists if C aI is nonzero as in holographic QFTs [18]. (iii) Even though the currentis proportional to small numbers ε I , its magnitude is amplified by Dirac’s delta, which wouldenable experimental detection. Consequently, the anomalous current produces the nonzeroamplitude (11). It strongly indicates that the real-photon emission is induced at the interfacesof, say, doped materials, which is a significant consequence of the nonzero anomalous currentindependent of the external gauge fields.Next, we shall consider a second example with τ ( x ) = τ ( x ). This form is motivatedby cosmology (although we are focusing on three dimensions and not four dimensions); incosmology, assuming the homogeneous isotropic universe, one usually considers a metric ds = dt − a ( t ) δ ij dx i dx j , where t is the cosmic time, and a ( t ) expresses the size of the uni-verse. Via a coordinate transformation dt = a [ t ( η )] dη, the metric reduces to the conformallyflat one: ds = a [ t ( η )] ( dη − δ ij dx i dx j ) ( η is called the conformal time). Identifying η = x ,we have τ ( x ) = ln a [ t ( x )]. We cannot apply the setup to the bulk but to the boundary of the observable universe as follows; The (spatial) boundary of our observable universecan be modeled by a 2-sphere with radius r universe . Since its curvature scaling as 1 /r is negligible, it can be approximated by the flat 2-space. Therefore, the boundary of theobservable universe (approximately) has a conformally flat metric, and our setup is applica-ble. The three-dimensional space-time has the Hubble parameter H d ( t ) := a − ( t ) da ( t ) /dt ,which characterizes the expansion of our universe. The three-dimensional Hubble parameternaturally enters in the derivative of τ : dτ ( x ) dx = H d [ t ( x )] a [ t ( x )] . (13)Now, let us turn to the current. Employing τ = τ ( x ), one obtains formulas; however,its physical meaning is unclear. To draw physical meaning, we take a reference time t = t ,and set a ( t ) = 1. Typically, t can be taken as the current age of the universe, whichis “long” in the ordinary sense. We consider a small deviation from the reference time, | ( t − t ) /t | (cid:28)
1. Then we have H d [ t ( x )] = ˙ a ( t ) + O [ | ( t ( x ) − t ) /t | ]. The leading term isthe three-dimensional Hubble constant: ˙ a ( t ) = ( H d ) . Expanding our currents in powers7f | ( t ( x ) − t ) /t | , we obtain (cid:104) j a ( x ) (cid:105) τλ = O [ | ( t ( x ) − t ) /t | ] , (cid:104) j a ( x ) (cid:105) τλ = 2 ∂ λ I ( x ) C aI [ λ ( x )] ( H d ) + O {| ( t ( x ) − t ) /t |} , (cid:104) j a ( x ) (cid:105) τλ = − ∂ λ I ( x ) C aI [ λ ( x )] ( H d ) + O {| ( t ( x ) − t ) /t |} . (14)This result implies that the anomalous current is possibly triggered by the expansion of theuniverse, which causes the photon emission from the edge of the observable universe. So,it is expected that the emitted photons mix with other sources of radiation such as cosmicmicrowave background (CMB). To see this, let us consider another set of background fields.The final example is λ I ( x ) = ε I θ ( x ) θ ( l − x ) and τ ( x ) = τ ( x ). Then we immediatelyobtain (cid:104) j a ( x ) (cid:105) τλ = 0 , (cid:104) j a ( x ) (cid:105) τλ = 0 , (cid:104) j a ( x ) (cid:105) τλ = − ε I C aI ( ε/ da [ t ( x )] dt ( x ) (cid:2) δ ( x ) − δ ( l − x ) (cid:3) . (15)In order to relate this result to cosmology, one can take, for example, particles on the bound-ary of our observable universe as sources of photon emission. Such particles, modeled by a3-ball, have one-dimensional (spatial) interfaces when projected to the boundary. Then withsuitable modifications on λ I ( x ), our result (15) predicts currents generated at the interfaces.Accordingly, photons are emitted at the interfaces. The photons mix with CMB, and thescattered sources of photons may explain fluctuations in CMB. It is intriguing to push thisconsideration forward. Acknowledgment
We are grateful to Xu-Guang Huang and Shinya Matsuzaki for useful comments.
References [1] G. ’t Hooft, “Naturalness, chiral symmetry, and spontaneous chiral symmetry breaking,”NATO Sci. Ser. B , 135 (1980). doi:10.1007/978-1-4684-7571-5 9[2] Z. Komargodski and A. Schwimmer, “On Renormalization Group Flows in Four Dimen-sions,” JHEP , 099 (2011) doi:10.1007/JHEP12(2011)099 [arXiv:1107.3987 [hep-th]]; Z. Komargodski, “The Constraints of Conformal Symmetry on RG Flows,” JHEP , 069 (2012) doi:10.1007/JHEP07(2012)069 [arXiv:1112.4538 [hep-th]].[3] D. T. Son and A. R. Zhitnitsky, Phys. Rev. D , 074018 (2004)doi:10.1103/PhysRevD.70.074018 [hep-ph/0405216].84] M. A. Metlitski and A. R. Zhitnitsky, Phys. Rev. D , 045011 (2005)doi:10.1103/PhysRevD.72.045011 [hep-ph/0505072].[5] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A , 227 (2008)doi:10.1016/j.nuclphysa.2008.02.298 [arXiv:0711.0950 [hep-ph]].[6] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D , 074033 (2008)doi:10.1103/PhysRevD.78.074033 [arXiv:0808.3382 [hep-ph]].[7] A. Vilenkin, Phys. Rev. D , 1807 (1979). doi:10.1103/PhysRevD.20.1807[8] A. Vilenkin, Phys. Rev. D , 2260 (1980). doi:10.1103/PhysRevD.21.2260[9] D. T. Son and P. Surowka, Phys. Rev. Lett. , 191601 (2009)doi:10.1103/PhysRevLett.103.191601 [arXiv:0906.5044 [hep-th]].[10] M. N. Chernodub, “Anomalous Transport Due to the Conformal Anomaly,” Phys. Rev.Lett. , no. 14, 141601 (2016) doi:10.1103/PhysRevLett.117.141601 [arXiv:1603.07993[hep-th]].[11] One can repeat our analysis below also in other dimensions including d = 4. The onlycondition which has to be satisfied in a space-time dimension one is interested in wouldbe that the trace anomaly in that dimension have terms quadratic in background fieldsso that the anomaly induces an anomalous transport to the second order. Therefore wecould perform our analysis in other dimensions but we focus on d = 3 for several reasons.Firstly, three-dimensional trace anomaly has fewer terms, making the analysis simpler.Secondly, it turns out that we can experimentally address the longstanding problem ofan existence of three-dimensional trace anomaly.[12] I. T. Drummond and G. M. Shore, “Conformal Anomalies for Interacting Scalar Fieldsin Curved Space-Time,” Phys. Rev. D , 1134 (1979). doi:10.1103/PhysRevD.19.1134;H. Osborn, “Renormalization and Composite Operators in Nonlinear σ Models,” Nucl.Phys. B , 595 (1987). doi:10.1016/0550-3213(87)90599-2[13] I. Jack and H. Osborn, “Constraints on RG Flow for Four Dimensional QuantumField Theories,” Nucl. Phys. B , 425 (2014) doi:10.1016/j.nuclphysb.2014.03.018[arXiv:1312.0428 [hep-th]]; G. M. Shore, “The c and a-theorems and the Local Renor-malisation Group,” doi:10.1007/978-3-319-54000-9 arXiv:1601.06662 [hep-th].[14] Since we consider responses to variation of background fields up to their quadraticorder, “seagull terms” abbreviated by ( · · · ) may also contribute. Here, we abused theterminology “seagull term,” and meant all quadratic terms in background fields notjust gauge fields A but also scalar fields τ and λ . So they may include not just Chern-Simons terms (cid:82) tr AdA + · · · , but also terms such as (cid:82) d xτ ( x ) λ I ( x ) J I ( x ) with somelocal operators J I . We discuss whether these terms contribute or not later.915] D. M. Capper and M. J. Duff, “Trace anomalies in dimensional regularization,” NuovoCim. A , 173 (1974). doi:10.1007/BF02748300; H. S. Tsao, “Conformal Anoma-lies in a General Background Metric,” Phys. Lett. , 79 (1977). doi:10.1016/0370-2693(77)90039-9; S. Deser, M. J. Duff and C. J. Isham, “Nonlocal Conformal Anoma-lies,” Nucl. Phys. B , 45 (1976). doi:10.1016/0550-3213(76)90480-6; M. J. Duff, “Ob-servations on Conformal Anomalies,” Nucl. Phys. B , 334 (1977). doi:10.1016/0550-3213(77)90410-2; S. Deser and A. Schwimmer, “Geometric classification of conformalanomalies in arbitrary dimensions,” Phys. Lett. B , 279 (1993) doi:10.1016/0370-2693(93)90934-A [hep-th/9302047].[16] Y. Nakayama, “Consistency of local renormalization group in d=3,” Nucl. Phys. B ,37 (2014) doi:10.1016/j.nuclphysb.2013.12.002 [arXiv:1307.8048 [hep-th]].[17] J. Wess and B. Zumino, “Consequences of anomalous Ward identities,” Phys. Lett. ,95 (1971). doi:10.1016/0370-2693(71)90582-X[18] K. Kikuchi, H. Hosoda and A. Suzuki, “On three-dimensional trace anomaly fromholographic local RG,” PTEP , no. 1, 013B02 (2017) doi:10.1093/ptep/ptw174[arXiv:1606.03979 [hep-th]].[19] Actually, the current can be found even when one relaxes the assumption of conformalsymmetry. In that case, the right-hand side of (7) also contains contributions fromoperators, but if the terms consist of just background fields that cannot be tuned awaywith local counterterms, one can still repeat our computation leading to (10). We areassuming conformal symmetry to relate our results on a conformally flat space (4) tothose on flat space-time; however, if one gives up translating the results back to flatspace-time, one does not have to assume the symmetry.[20] In QCD and QED, quantum anomalies of the scale symmetry or the chiral symmetryinduce the anomalous transports depending on background gauge fields. If one takes ex-plicit scale/chiral symmetry breaking into account, the nonzero fermion mass also givesadditional contributions to the trace of the energy-momentum tensor or divergence of thecurrent, respectively. So, one might deduce that the nonzero fermion mass could induceanomalous transports independent of background gauge fields, which may be derivedfrom a correlation function like (cid:104) j µ ( x ) ¯ ψψ ( y ) (cid:105) ; the one-point function (cid:104) ¯ ψψ ( y ) (cid:105) can belinear in background gauge fields, resulting in additional contributions to (cid:104) j µ ( x ) ¯ ψψ ( y ) (cid:105) independent of background gauge fields. However, the correlator (cid:104) j µ ( x ) ¯ ψψ ( y ) (cid:105) vanishesin QCD and QED due to their (exact) charge conjugation symmetry. Since all contribu-tions proportional to background gauge fields one obtains by bringing them down fromthe action vanish in the process A →
0, one concludes that in QCD and QED one cannotfind nonzero anomalous transports in the absence of background gauge fields.1021] Strictly speaking, since there is no conformal symmetry, (7) is no longer an anomaly.However, we continue to simply call the current induced by (7) as anomalous currentfollowing [10] because we are assuming it is physical.[22] Y. Kim, I. J. Shin and T. Tsukioka, “Holographic QCD: Past, Present, and Fu-ture,” Prog. Part. Nucl. Phys. , 55-112 (2013) doi:10.1016/j.ppnp.2012.09.002[arXiv:1205.4852 [hep-ph]].[23] K. Fukushima and K. Mameda, Phys. Rev. D86